Hmm, I've just posted (my first post) in the 1st gen Cooper forum about a similar issue I see with a car we may be buying. Into reverse is fine but into 'D' it stalls. The only way I get it moving is left foot braking and right foot giving it some throttle as I move the lever into Drive.

So no particular help from me but am curious about any possible solutions too.

1) Sudden idle RPM drop on Cooper CVT models during initial 5 minutes of driving when engine is cold, regardless of ambient temperature. (Upcoming fix CD #33?) Car often experiences a near stall/or complete engine stall condition. Current workaround fix is to put key in ignition and turn to "ON" position. Hold the key there for 10 to 20 seconds (Allowing DME to initialize) and then start engine normally. This seems to virtually eliminate the near stall problem when engine is cold. Some regular 5-speed Coopers have also reported this problem.

Two suggestions from experience:
1. Make sure your car battery is in excellent order.
2. A small blip on the accelerator (only a slight tap, say, to 1,200 rpm) just after starting and before putting into D or R seems to eliminate the engine hesitation.